Trig Identities

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How to Use the Double-Angle Identity for Sine

The double-angle formula for sine comes from using the trig identity for the sine of a sum, sin (α + β) = sinαcosβ + cosαsinβ. If α = β, then you can replace β with α in the formula, giving you [more…]

Pythagorean Sine and Cosine Identities on a Unit Circle

If you have ever wondered why the Pythagorean identity, sin2θ + cos2θ = 1, is so important, and where it came from, then read on. This identity is important because it sets an expression involving trig [more…]

Cotangent and Cosecant Identities on a Unit Circle

Starting with the Pythagorean identity, sin2θ + cos2θ = 1, you can derive cotangent and cosecant Pythagorean identities. All you do is throw in a little algebra and apply the reciprocal and ratio identities [more…]

Rearrange the Pythagorean Identities

Familiarizing yourself with the different versions of Pythagorean identities is helpful so that you can easily recognize them when solving trigonometry equations or simplifying expressions. [more…]

Express Sine in Terms of Cosine

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you [more…]

Express Sine in Terms of Tangent

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you [more…]

Express Sine in Terms of Cotangent

Even though each trigonometry function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you [more…]

Reciprocal Trigonometry Identities

The simplest and most basic trig identities (equations of equivalence) are those involving the reciprocals of the trigonometry functions. To jog your memory, a reciprocal of a number is 1 divided by that [more…]

Tangent and Secant Identities on a Unit Circle

Starting with the Pythagorean identity, sin2θ + cos2θ = 1, you can derive tangent and secant Pythagorean identities. All you do is throw in a little algebra and apply the reciprocal and ratio identities [more…]

How to Use the Angle-Sum Identity When You Don’t Know the Angle

In some trigonometry problems, you may not know what the measure of an angle is, but you know something about the angle’s function values. For example, suppose you have two angles, α in the second quadrant [more…]

How to Use the Subtraction Identities in a Trig Problem

You can find function values of angles using angle-addition identities. And you have more possibilities for finding the function values of angles when you use subtraction in a trig problem. For example [more…]

Find Trigonometry Ratio Identities

Trig has two identities called ratio identities. This label can be confusing, because all the trig functions are defined by ratios. Somewhere along the line, however, mathematicians thought this description [more…]

Basic Pythagorean Identities for Trigonometry Functions

The Pythagorean identities are building blocks for many of the manipulations of trigonometric equations and expressions. They provide a greater number of methods for solving trig problems more efficiently [more…]

Express Sine in Terms of Secant or Cosecant

Even though each trig function is perfectly wonderful, being able to express each trig function in terms of one of the other five trig functions is frequently to your advantage. For example, you may have [more…]

The Origin of the Half-Angle Identities for Sine

The trig identities come in sums, differences, ratios, multiples, and halves. With a half-angle identity, you can get the value of a sine for a 15-degree angle using a function of of 30 degree angle. You [more…]

How to Find Half-Angle Identities for Tangent

The half-angle trig identity for tangent has two versions. Rather than this being a nuisance, having more than one option is really rather nice, because you can choose the version that works best for your [more…]

Dealing with Half-Angle Identities Involving Radicals

By adding, subtracting, or doubling angle measures, you can find lots of exact values of trigonometry functions. For example, you can use the half-angle identity when the exact value of the trig function [more…]

How to Find a Common Denominator of a Fraction to Solve a Trig Identity

Fractions are your friends. You may not believe this now, but the more you work with trigonometry functions, the more you’ll like fractions. Finding a common denominator to combine fractions often paves [more…]

How to Multiply by a Conjugate to Find a Trigonometry Identity

Conjugates offer a great way to find trigonometry identities. In mathematics, a conjugate consists of the same two terms as the first expression, separated by the opposite sign. For instance, the conjugate [more…]

How to Square Both Sides to Solve a Trigonometry Identity Problem

When you work on both sides of a trig identity at the same time, you may sometimes need to square both sides. You generally use this technique when one side or the other [more…]

Using the Angle-Sum Identity

Three basic trigonometry identities involve the sums of angles; the functions involved in these identities are sine, cosine, and tangent. You can also adapt these three basic angle-sum identities for the [more…]

Using the Double-Angle Identity for Cosine

Identities for angles that are twice as large as one of the common angles (double angles) are used frequently in trig. These identities allow you to deal with a larger angle in the terms of a smaller and [more…]

How to Use Half-Angle Identities to Find the Sine of an Angle

By adding, subtracting, or doubling angle measures, you can find lots of exact values of trigonometry functions using the functions of angles you already know. For example, even though you can use a difference [more…]

How to Work Both Sides of a Trig Identity

With a trigonometry identity, working on both sides of the equation is even more fun than working on both sides of an algebraic equation. In algebra, you can multiply each side by the same number, square [more…]

Change to Sines and Cosines in a Trigonometry Identity

With some trig identities, you may decide to simplify matters by either changing everything to sines and cosines or by factoring out some function. Sometimes, it isn’t clear which side you should work [more…]

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